<h2>Problem 136</h2>
<div style="color:#666;font-size:80%;">29 December 2006</div><br />
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<p>The positive integers, <i>x</i>, <i>y</i>, and <i>z</i>, are consecutive terms of an arithmetic progression. Given that <i>n</i> is a positive integer, the equation, <i>x</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> <i>y</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> <i>z</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = <i>n</i>, has exactly one solution when <i>n</i> = 20:</p>
<p style='text-align:center;'>13<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> 10<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> 7<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 20</p>
<p>In fact there are twenty-five values of <i>n</i> below one hundred for which the equation has a unique solution.</p>
<p>How many values of <i>n</i> less than fifty million have exactly one solution?</p>

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